Despite being very generic, versatile, and widely used in many high-order applications,
there are a few aspects of the Lorentz tensor decomposition approach that makes the traditional projection method
not so easy to be carried out in certain cases, as will be discussed in the next section.
For example, besides facing complexities in deriving D-dimensional projectors for tensor
decomposition coefficients in some multiple-parton, multiple-scale scattering processes,
evanescent Lorentz structuresThe evanescent Lorentz structures appearing in a Lorentz tensor decomposition should not be confused with operator mixings in the renormalization of composite operators in effective field theories [1]}, [2]}, nor with evanescent terms in the DRED or FDH regularized Lagrangian [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}. can appear in the D-dimensional basis for the loop amplitudes in question.
Their presence can lead to intermediate spurious poles in the resulting D-dimensional projectors [13]}, [14]}, [15]}.
Furthermore, when there are several external fermions involved in the scattering [14]}, [17]},
the complete and linearly independent set of basis
structures in D dimensions will generally increase with the perturbative order at which the
virtual amplitude is computed (as the Dirac algebra is formally infinite-dimensional in non-integer D dimensions).
